Finite-function collapse to countable size ($\mathrm{Col}(\omega ,\kappa )$)

For an infinite ground-model cardinal number $\kappa$, use finite partial functions from $\omega$ to $\kappa$, ordered by reverse inclusion. The union of a generic filter is a surjection $\omega \to \kappa$. The order has size $\kappa$, hence the ${\kappa }^{+}$-chain condition, and preserves cardinals at least ${\kappa }^{+}$.

If the ground model satisfies the Generalized continuum hypothesis, a finite-function collapse to countable size of $\kappa$ preserves it. The old ${\kappa }^{+}$ becomes the new ${\aleph }_1$, and there are at most $2^{\kappa }={\kappa }^{+}$ names for subsets of $\omega$. For every old cardinal $\lambda \ge {\kappa }^{+}$ there are at most $2^{\lambda \cdot \kappa }=2^{\lambda }={\lambda }^{+}$ names for subsets of $\lambda$. Cardinal preservation by chain-condition forcing and Cantor theorem turn these upper bounds into the required equalities.